Question

If $\int\limits_{\frac{1}{3}}^{3}\left|\log _e x\right| d x=\frac{m}{n} \log e\left(\frac{n^2}{v}\right)$, where $m$ and $n$ are coprime natural numbers, then $m^2+n^2-5$ is equal to _____

Answer 1

The correct answer is 20.
31​∫3​∣ℓnx∣dx=31​∫1​(−ℓnx)dx+1∫3​(ℓnx)dx
=−[xℓnx−x]1/31​+[xℓnx−x]13​
=−[−1−(31​ℓn31​−31​)]+[3ℓn3−3−(−1)]
=[−32​−31​ℓn31​]+[3ℓn3−2]
=−34​+38​ℓn3
=34​(2ℓn3−1)
=34​(ℓne9​)
∴m=4,n=3
Now, m2+n2−5=16+9−5=20